There are two parts to this question.
The first part seems pretty straightforward except how anything can be
true and false at the same time seems like a big no-no in logic.
I agree with the answer provided by Geoffrey Thomas. This is only an alternate way to explain it.
Consider the following proof. Think of "P" in the proof as "It is raining." Think of not "P", or "¬P", in the proof as "It is not raining."
In order to show that one can get anything from a contradiction, we assume we have a contradiction. That is what is happening on line 1. Both "It is raining" and "It is not raining" or "P ∧ ¬P" we assume to be true.
What follows from assuming we have a contradiction on line 1? The claim is that anything should follow. How do we represent anything? Just give it a name and don't specify it. Here I am saying "Q" will stand for "anything". I want to derive anything, or "Q", from the contradiction in line 1, "P ∧ ¬P".
The steps show how this is done.
On lines 2 and 3, I split out the two sentences connected by "and" (∧). On line 2, I put "P". On line 3, I put "¬P". The two sentences on either side of an "and" sentence are called conjuncts and the full sentence is called a conjunction. When I take the two conjuncts and put them on their own lines, I eliminate the conjunction. This is written with the rule "∧E" and they both refer to line 1.
On line 4, I take the "P" on line 2 and realize that if "P" is true it doesn't matter what I "or" with "P". It could be anything. I have a symbol for anything. I called it "Q", so I write "P" or "Q" on line 4 as "P ∨ Q". This or-ing process is called a disjunction and I have just introduced a disjunction on line 4, so I write the rule "∨I" and reference line 2 since that is the line "P" is on.
On line 5, I get the result I am looking for, because I don't have only "P", I also have "¬P" from line 3 which came from the contradiction I assumed on line 1. That means "P" is false (if I look at it from the perspective of "¬P"). Forget that on line 2 I thought "P" was true. This is a contradiction we are working with. So if "P" is false and "P ∨ Q" is true, that means "Q" is true. But "Q" was anything. The rule for doing this is called disjunctive syllogism (DS).
That means if I have a contradiction, I can derive anything.
The second quoted part about why contradiction "should" imply
everything I can't really follow. Which is the preorder and on what
propositions? Isn't it arbitrary to come up with the statement "P is
less then or equal to Q? And why is contradiction minimum in this
I agree with the answer provided by user26652 for this part, but I will try to describe it differently.
Note that Stefan Perko is not trying to argue that having a contradiction implies anything. He's already done that in the first part and he assumes you have accepted it.
Rather he is trying to show that there is an intuitive way of thinking about a contradiction from the perspective of set theory by thinking of a contradiction as something insignificant, even empty, like the empty set. The empty set is a subset of any set (that is, it is the subset of "anything") just like a contradiction implies anything. He is pointing out a similarity between a contradiction and the empty set.
It seems odd to think of a contradiction as something small. There is even a rule called explosion. If we get a contradiction we can use the rule of explosion and derive anything we want as the first part showed.
However, to see how smallness might be a good metaphor for a contradiction rather than a large explosion consider what an implication might look like using Venn diagrams. In the following diagram "P" implies "Q". Whenever we have "P", we have "Q". "Q" is in a sense bigger than "P". Note that "P" represents a possibly smaller circle than "Q" in the diagram. It cannot be a circle bigger than "Q".
Now think of "P" as a contradiction getting smaller and smaller in that diagram and implying anything. Just like the empty set is so small that it is a subset of any set.